\(\int \frac {1}{(b d+2 c d x) (a+b x+c x^2)^{5/2}} \, dx\) [1256]

Optimal result

Integrand size = 26, antiderivative size = 118 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {8 c}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}+\frac {16 c^{3/2} \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} d} \]

[Out]

-2/3/(-4*a*c+b^2)/d/(c*x^2+b*x+a)^(3/2)+16*c^(3/2)*arctan(2*c^(1/2)*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^(1/2))/(-
4*a*c+b^2)^(5/2)/d+8*c/(-4*a*c+b^2)^2/d/(c*x^2+b*x+a)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {701, 702, 211} \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {16 c^{3/2} \arctan \left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{d \left (b^2-4 a c\right )^{5/2}}+\frac {8 c}{d \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2}{3 d \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]

[In]

Int[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

-2/(3*(b^2 - 4*a*c)*d*(a + b*x + c*x^2)^(3/2)) + (8*c)/((b^2 - 4*a*c)^2*d*Sqrt[a + b*x + c*x^2]) + (16*c^(3/2)
*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c)^(5/2)*d)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 701

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2*c*(d + e*x)^(m + 1
)*((a + b*x + c*x^2)^(p + 1)/(e*(p + 1)*(b^2 - 4*a*c))), x] - Dist[2*c*e*((m + 2*p + 3)/(e*(p + 1)*(b^2 - 4*a*
c))), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0] && NeQ[m + 2*p + 3, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && RationalQ[m] && IntegerQ[2*p]

Rule 702

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[4*c, Subst[Int[1/(b^2*e
- 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}-\frac {(4 c) \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{3/2}} \, dx}{b^2-4 a c} \\ & = -\frac {2}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {8 c}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}+\frac {\left (16 c^2\right ) \int \frac {1}{(b d+2 c d x) \sqrt {a+b x+c x^2}} \, dx}{\left (b^2-4 a c\right )^2} \\ & = -\frac {2}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {8 c}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}+\frac {\left (64 c^3\right ) \text {Subst}\left (\int \frac {1}{2 b^2 c d-8 a c^2 d+8 c^2 d x^2} \, dx,x,\sqrt {a+b x+c x^2}\right )}{\left (b^2-4 a c\right )^2} \\ & = -\frac {2}{3 \left (b^2-4 a c\right ) d \left (a+b x+c x^2\right )^{3/2}}+\frac {8 c}{\left (b^2-4 a c\right )^2 d \sqrt {a+b x+c x^2}}+\frac {16 c^{3/2} \tan ^{-1}\left (\frac {2 \sqrt {c} \sqrt {a+b x+c x^2}}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2} d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1624\) vs. \(2(118)=236\).

Time = 5.78 (sec) , antiderivative size = 1624, normalized size of antiderivative = 13.76 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \left (-8192 a^{15/2} \left (b^2+12 b c x+12 c^2 x^2\right )-\sqrt {a} b^2 x^7 (b+c x)^4 \left (b^3+212 b^2 c x+308 b c^2 x^2+96 c^3 x^3\right )-2048 a^{13/2} x \left (14 b^3+203 b^2 c x+348 b c^2 x^2+156 c^3 x^3\right )-512 a^{11/2} x^2 \left (77 b^4+1362 b^3 c x+3272 b^2 c^2 x^2+2760 b c^3 x^3+776 c^4 x^4\right )-2 a^{3/2} x^6 (b+c x)^3 \left (49 b^5+3251 b^4 c x+6878 b^3 c^2 x^2+4740 b^2 c^3 x^3+1144 b c^4 x^4+64 c^5 x^5\right )-8 a^{5/2} x^5 (b+c x)^2 \left (196 b^5+7609 b^4 c x+19480 b^3 c^2 x^2+18023 b^2 c^3 x^3+6836 b c^4 x^4+860 c^5 x^5\right )-128 a^{9/2} x^3 \left (210 b^5+4636 b^4 c x+14248 b^3 c^2 x^2+17237 b^2 c^3 x^3+9244 b c^4 x^4+1828 c^5 x^5\right )-32 a^{7/2} x^4 \left (294 b^6+8464 b^5 c x+31565 b^4 c^2 x^2+48878 b^3 c^3 x^3+37475 b^2 c^4 x^4+14012 b c^5 x^5+2020 c^6 x^6\right )+16 b^3 c x^8 (b+c x)^5 \sqrt {a+x (b+c x)}+8192 a^7 \left (b^2+12 b c x+12 c^2 x^2\right ) \sqrt {a+x (b+c x)}+6144 a^6 x \left (4 b^3+59 b^2 c x+100 b c^2 x^2+44 c^3 x^3\right ) \sqrt {a+x (b+c x)}+2 a b x^6 (b+c x)^3 \left (7 b^4+711 b^3 c x+1296 b^2 c^2 x^2+684 b c^3 x^3+96 c^4 x^4\right ) \sqrt {a+x (b+c x)}+512 a^5 x^2 \left (55 b^4+1012 b^3 c x+2392 b^2 c^2 x^2+1968 b c^3 x^3+536 c^4 x^4\right ) \sqrt {a+x (b+c x)}+8 a^2 x^5 (b+c x)^2 \left (56 b^5+2695 b^4 c x+6406 b^3 c^2 x^2+5293 b^2 c^3 x^3+1700 b c^4 x^4+164 c^5 x^5\right ) \sqrt {a+x (b+c x)}+128 a^4 x^3 \left (120 b^5+2844 b^4 c x+8568 b^3 c^2 x^2+10043 b^2 c^3 x^3+5172 b c^4 x^4+972 c^5 x^5\right ) \sqrt {a+x (b+c x)}+32 a^3 x^4 \left (126 b^6+4088 b^5 c x+14875 b^4 c^2 x^2+22124 b^3 c^3 x^3+16091 b^2 c^4 x^4+5620 b c^5 x^5+740 c^6 x^6\right ) \sqrt {a+x (b+c x)}\right )}{3 \sqrt {a} \left (b^2-4 a c\right )^2 d (a+x (b+c x)) \left (8192 a^{15/2}+2 \sqrt {a} b^2 x^7 (b+c x)^4 (7 b+3 c x)+2048 a^{13/2} x (16 b+13 c x)+512 a^{11/2} x^2 \left (103 b^2+164 b c x+64 c^2 x^2\right )+2 a^{3/2} x^6 (b+c x)^3 \left (231 b^3+255 b^2 c x+72 b c^2 x^2+4 c^3 x^3\right )+8 a^{5/2} x^5 (b+c x)^2 \left (560 b^3+903 b^2 c x+438 b c^2 x^2+61 c^3 x^3\right )+128 a^{9/2} x^3 \left (340 b^3+792 b^2 c x+600 b c^2 x^2+147 c^3 x^3\right )+32 a^{7/2} x^4 \left (606 b^4+1824 b^3 c x+1995 b^2 c^2 x^2+932 b c^3 x^3+155 c^4 x^4\right )-8192 a^7 \sqrt {a+x (b+c x)}-b^3 x^7 (b+c x)^4 \sqrt {a+x (b+c x)}-2048 a^6 x (14 b+11 c x) \sqrt {a+x (b+c x)}-2 a b x^6 (b+c x)^3 \left (49 b^2+39 b c x+6 c^2 x^2\right ) \sqrt {a+x (b+c x)}-512 a^5 x^2 \left (77 b^2+118 b c x+44 c^2 x^2\right ) \sqrt {a+x (b+c x)}-8 a^2 x^5 (b+c x)^2 \left (196 b^3+273 b^2 c x+108 b c^2 x^2+11 c^3 x^3\right ) \sqrt {a+x (b+c x)}-128 a^4 x^3 \left (210 b^3+468 b^2 c x+336 b c^2 x^2+77 c^3 x^3\right ) \sqrt {a+x (b+c x)}-32 a^3 x^4 \left (294 b^4+840 b^3 c x+861 b^2 c^2 x^2+370 b c^3 x^3+55 c^4 x^4\right ) \sqrt {a+x (b+c x)}\right )}-\frac {32 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c} \sqrt {-b^2+4 a c} x}{\sqrt {a} (b+2 c x)-b \sqrt {a+x (b+c x)}}\right )}{\left (-b^2+4 a c\right )^{5/2} d} \]

[In]

Integrate[1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(5/2)),x]

[Out]

(-2*(-8192*a^(15/2)*(b^2 + 12*b*c*x + 12*c^2*x^2) - Sqrt[a]*b^2*x^7*(b + c*x)^4*(b^3 + 212*b^2*c*x + 308*b*c^2
*x^2 + 96*c^3*x^3) - 2048*a^(13/2)*x*(14*b^3 + 203*b^2*c*x + 348*b*c^2*x^2 + 156*c^3*x^3) - 512*a^(11/2)*x^2*(
77*b^4 + 1362*b^3*c*x + 3272*b^2*c^2*x^2 + 2760*b*c^3*x^3 + 776*c^4*x^4) - 2*a^(3/2)*x^6*(b + c*x)^3*(49*b^5 +
3251*b^4*c*x + 6878*b^3*c^2*x^2 + 4740*b^2*c^3*x^3 + 1144*b*c^4*x^4 + 64*c^5*x^5) - 8*a^(5/2)*x^5*(b + c*x)^2
*(196*b^5 + 7609*b^4*c*x + 19480*b^3*c^2*x^2 + 18023*b^2*c^3*x^3 + 6836*b*c^4*x^4 + 860*c^5*x^5) - 128*a^(9/2)
*x^3*(210*b^5 + 4636*b^4*c*x + 14248*b^3*c^2*x^2 + 17237*b^2*c^3*x^3 + 9244*b*c^4*x^4 + 1828*c^5*x^5) - 32*a^(
7/2)*x^4*(294*b^6 + 8464*b^5*c*x + 31565*b^4*c^2*x^2 + 48878*b^3*c^3*x^3 + 37475*b^2*c^4*x^4 + 14012*b*c^5*x^5
+ 2020*c^6*x^6) + 16*b^3*c*x^8*(b + c*x)^5*Sqrt[a + x*(b + c*x)] + 8192*a^7*(b^2 + 12*b*c*x + 12*c^2*x^2)*Sqr
t[a + x*(b + c*x)] + 6144*a^6*x*(4*b^3 + 59*b^2*c*x + 100*b*c^2*x^2 + 44*c^3*x^3)*Sqrt[a + x*(b + c*x)] + 2*a*
b*x^6*(b + c*x)^3*(7*b^4 + 711*b^3*c*x + 1296*b^2*c^2*x^2 + 684*b*c^3*x^3 + 96*c^4*x^4)*Sqrt[a + x*(b + c*x)]
+ 512*a^5*x^2*(55*b^4 + 1012*b^3*c*x + 2392*b^2*c^2*x^2 + 1968*b*c^3*x^3 + 536*c^4*x^4)*Sqrt[a + x*(b + c*x)]
+ 8*a^2*x^5*(b + c*x)^2*(56*b^5 + 2695*b^4*c*x + 6406*b^3*c^2*x^2 + 5293*b^2*c^3*x^3 + 1700*b*c^4*x^4 + 164*c^
5*x^5)*Sqrt[a + x*(b + c*x)] + 128*a^4*x^3*(120*b^5 + 2844*b^4*c*x + 8568*b^3*c^2*x^2 + 10043*b^2*c^3*x^3 + 51
72*b*c^4*x^4 + 972*c^5*x^5)*Sqrt[a + x*(b + c*x)] + 32*a^3*x^4*(126*b^6 + 4088*b^5*c*x + 14875*b^4*c^2*x^2 + 2
2124*b^3*c^3*x^3 + 16091*b^2*c^4*x^4 + 5620*b*c^5*x^5 + 740*c^6*x^6)*Sqrt[a + x*(b + c*x)]))/(3*Sqrt[a]*(b^2 -
4*a*c)^2*d*(a + x*(b + c*x))*(8192*a^(15/2) + 2*Sqrt[a]*b^2*x^7*(b + c*x)^4*(7*b + 3*c*x) + 2048*a^(13/2)*x*(
16*b + 13*c*x) + 512*a^(11/2)*x^2*(103*b^2 + 164*b*c*x + 64*c^2*x^2) + 2*a^(3/2)*x^6*(b + c*x)^3*(231*b^3 + 25
5*b^2*c*x + 72*b*c^2*x^2 + 4*c^3*x^3) + 8*a^(5/2)*x^5*(b + c*x)^2*(560*b^3 + 903*b^2*c*x + 438*b*c^2*x^2 + 61*
c^3*x^3) + 128*a^(9/2)*x^3*(340*b^3 + 792*b^2*c*x + 600*b*c^2*x^2 + 147*c^3*x^3) + 32*a^(7/2)*x^4*(606*b^4 + 1
824*b^3*c*x + 1995*b^2*c^2*x^2 + 932*b*c^3*x^3 + 155*c^4*x^4) - 8192*a^7*Sqrt[a + x*(b + c*x)] - b^3*x^7*(b +
c*x)^4*Sqrt[a + x*(b + c*x)] - 2048*a^6*x*(14*b + 11*c*x)*Sqrt[a + x*(b + c*x)] - 2*a*b*x^6*(b + c*x)^3*(49*b^
2 + 39*b*c*x + 6*c^2*x^2)*Sqrt[a + x*(b + c*x)] - 512*a^5*x^2*(77*b^2 + 118*b*c*x + 44*c^2*x^2)*Sqrt[a + x*(b
+ c*x)] - 8*a^2*x^5*(b + c*x)^2*(196*b^3 + 273*b^2*c*x + 108*b*c^2*x^2 + 11*c^3*x^3)*Sqrt[a + x*(b + c*x)] - 1
28*a^4*x^3*(210*b^3 + 468*b^2*c*x + 336*b*c^2*x^2 + 77*c^3*x^3)*Sqrt[a + x*(b + c*x)] - 32*a^3*x^4*(294*b^4 +
840*b^3*c*x + 861*b^2*c^2*x^2 + 370*b*c^3*x^3 + 55*c^4*x^4)*Sqrt[a + x*(b + c*x)])) - (32*c^(3/2)*ArcTanh[(Sqr
t[c]*Sqrt[-b^2 + 4*a*c]*x)/(Sqrt[a]*(b + 2*c*x) - b*Sqrt[a + x*(b + c*x)])])/((-b^2 + 4*a*c)^(5/2)*d)

Maple [A] (verified)

Time = 2.79 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.12

method	result	size
pseudoelliptic	\(-\frac {16 \left (c^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {2 c \sqrt {c \,x^{2}+b x +a}}{\sqrt {4 c^{2} a -b^{2} c}}\right )-\frac {2 \sqrt {4 c^{2} a -b^{2} c}\, \left (\frac {3 c^{2} x^{2}}{4}+\left (\frac {3 b x}{4}+a \right ) c -\frac {b^{2}}{16}\right )}{3}\right )}{\sqrt {4 c^{2} a -b^{2} c}\, \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (4 a c -b^{2}\right )^{2} d}\)	\(132\)
default	\(\frac {\frac {4 c}{3 \left (4 a c -b^{2}\right ) \left (\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}\right )^{\frac {3}{2}}}+\frac {4 c \left (\frac {4 c}{\left (4 a c -b^{2}\right ) \sqrt {\left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{4 c}}}-\frac {8 c \ln \left (\frac {\frac {4 a c -b^{2}}{2 c}+\frac {\sqrt {\frac {4 a c -b^{2}}{c}}\, \sqrt {4 \left (x +\frac {b}{2 c}\right )^{2} c +\frac {4 a c -b^{2}}{c}}}{2}}{x +\frac {b}{2 c}}\right )}{\left (4 a c -b^{2}\right ) \sqrt {\frac {4 a c -b^{2}}{c}}}\right )}{4 a c -b^{2}}}{2 d c}\)	\(223\)

[In]

int(1/(2*c*d*x+b*d)/(c*x^2+b*x+a)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-16*(c^2*(c*x^2+b*x+a)^(3/2)*arctanh(2*c*(c*x^2+b*x+a)^(1/2)/(4*a*c^2-b^2*c)^(1/2))-2/3*(4*a*c^2-b^2*c)^(1/2)*
(3/4*c^2*x^2+(3/4*b*x+a)*c-1/16*b^2))/(4*a*c^2-b^2*c)^(1/2)/(c*x^2+b*x+a)^(3/2)/(4*a*c-b^2)^2/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (102) = 204\).

Time = 0.57 (sec) , antiderivative size = 620, normalized size of antiderivative = 5.25 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\left [\frac {2 \, {\left (12 \, {\left (c^{3} x^{4} + 2 \, b c^{2} x^{3} + 2 \, a b c x + a^{2} c + {\left (b^{2} c + 2 \, a c^{2}\right )} x^{2}\right )} \sqrt {-\frac {c}{b^{2} - 4 \, a c}} \log \left (-\frac {4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c + 4 \, \sqrt {c x^{2} + b x + a} {\left (b^{2} - 4 \, a c\right )} \sqrt {-\frac {c}{b^{2} - 4 \, a c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right ) + {\left (12 \, c^{2} x^{2} + 12 \, b c x - b^{2} + 16 \, a c\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d x + {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} d\right )}}, \frac {2 \, {\left (24 \, {\left (c^{3} x^{4} + 2 \, b c^{2} x^{3} + 2 \, a b c x + a^{2} c + {\left (b^{2} c + 2 \, a c^{2}\right )} x^{2}\right )} \sqrt {\frac {c}{b^{2} - 4 \, a c}} \arctan \left (-\frac {\sqrt {c x^{2} + b x + a} {\left (b^{2} - 4 \, a c\right )} \sqrt {\frac {c}{b^{2} - 4 \, a c}}}{2 \, {\left (c^{2} x^{2} + b c x + a c\right )}}\right ) + {\left (12 \, c^{2} x^{2} + 12 \, b c x - b^{2} + 16 \, a c\right )} \sqrt {c x^{2} + b x + a}\right )}}{3 \, {\left ({\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} d x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} d x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} d x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} d x + {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2}\right )} d\right )}}\right ] \]

[In]

integrate(1/(2*c*d*x+b*d)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")

[Out]

[2/3*(12*(c^3*x^4 + 2*b*c^2*x^3 + 2*a*b*c*x + a^2*c + (b^2*c + 2*a*c^2)*x^2)*sqrt(-c/(b^2 - 4*a*c))*log(-(4*c^
2*x^2 + 4*b*c*x - b^2 + 8*a*c + 4*sqrt(c*x^2 + b*x + a)*(b^2 - 4*a*c)*sqrt(-c/(b^2 - 4*a*c)))/(4*c^2*x^2 + 4*b
*c*x + b^2)) + (12*c^2*x^2 + 12*b*c*x - b^2 + 16*a*c)*sqrt(c*x^2 + b*x + a))/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*
c^4)*d*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*d*x^2 + 2*(a*b^5 -
8*a^2*b^3*c + 16*a^3*b*c^2)*d*x + (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*d), 2/3*(24*(c^3*x^4 + 2*b*c^2*x^3 + 2*
a*b*c*x + a^2*c + (b^2*c + 2*a*c^2)*x^2)*sqrt(c/(b^2 - 4*a*c))*arctan(-1/2*sqrt(c*x^2 + b*x + a)*(b^2 - 4*a*c)
*sqrt(c/(b^2 - 4*a*c))/(c^2*x^2 + b*c*x + a*c)) + (12*c^2*x^2 + 12*b*c*x - b^2 + 16*a*c)*sqrt(c*x^2 + b*x + a)
)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*d*x^3 + (b^6 - 6*a*b^4*
c + 32*a^3*c^3)*d*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*d*x + (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*d)]

Sympy [F]

\[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {\int \frac {1}{a^{2} b \sqrt {a + b x + c x^{2}} + 2 a^{2} c x \sqrt {a + b x + c x^{2}} + 2 a b^{2} x \sqrt {a + b x + c x^{2}} + 6 a b c x^{2} \sqrt {a + b x + c x^{2}} + 4 a c^{2} x^{3} \sqrt {a + b x + c x^{2}} + b^{3} x^{2} \sqrt {a + b x + c x^{2}} + 4 b^{2} c x^{3} \sqrt {a + b x + c x^{2}} + 5 b c^{2} x^{4} \sqrt {a + b x + c x^{2}} + 2 c^{3} x^{5} \sqrt {a + b x + c x^{2}}}\, dx}{d} \]

[In]

integrate(1/(2*c*d*x+b*d)/(c*x**2+b*x+a)**(5/2),x)

[Out]

Integral(1/(a**2*b*sqrt(a + b*x + c*x**2) + 2*a**2*c*x*sqrt(a + b*x + c*x**2) + 2*a*b**2*x*sqrt(a + b*x + c*x*
*2) + 6*a*b*c*x**2*sqrt(a + b*x + c*x**2) + 4*a*c**2*x**3*sqrt(a + b*x + c*x**2) + b**3*x**2*sqrt(a + b*x + c*
x**2) + 4*b**2*c*x**3*sqrt(a + b*x + c*x**2) + 5*b*c**2*x**4*sqrt(a + b*x + c*x**2) + 2*c**3*x**5*sqrt(a + b*x
+ c*x**2)), x)/d

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate(1/(2*c*d*x+b*d)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more deta

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 906 vs. \(2 (102) = 204\).

Time = 0.31 (sec) , antiderivative size = 906, normalized size of antiderivative = 7.68 \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {32 \, c^{2} \arctan \left (-\frac {2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x + a}\right )} c + b \sqrt {c}}{\sqrt {b^{2} c - 4 \, a c^{2}}}\right )}{{\left (b^{4} d - 8 \, a b^{2} c d + 16 \, a^{2} c^{2} d\right )} \sqrt {b^{2} c - 4 \, a c^{2}}} + \frac {2 \, {\left (12 \, {\left (\frac {{\left (b^{16} c^{2} d^{3} - 32 \, a b^{14} c^{3} d^{3} + 448 \, a^{2} b^{12} c^{4} d^{3} - 3584 \, a^{3} b^{10} c^{5} d^{3} + 17920 \, a^{4} b^{8} c^{6} d^{3} - 57344 \, a^{5} b^{6} c^{7} d^{3} + 114688 \, a^{6} b^{4} c^{8} d^{3} - 131072 \, a^{7} b^{2} c^{9} d^{3} + 65536 \, a^{8} c^{10} d^{3}\right )} x}{b^{20} d^{4} - 40 \, a b^{18} c d^{4} + 720 \, a^{2} b^{16} c^{2} d^{4} - 7680 \, a^{3} b^{14} c^{3} d^{4} + 53760 \, a^{4} b^{12} c^{4} d^{4} - 258048 \, a^{5} b^{10} c^{5} d^{4} + 860160 \, a^{6} b^{8} c^{6} d^{4} - 1966080 \, a^{7} b^{6} c^{7} d^{4} + 2949120 \, a^{8} b^{4} c^{8} d^{4} - 2621440 \, a^{9} b^{2} c^{9} d^{4} + 1048576 \, a^{10} c^{10} d^{4}} + \frac {b^{17} c d^{3} - 32 \, a b^{15} c^{2} d^{3} + 448 \, a^{2} b^{13} c^{3} d^{3} - 3584 \, a^{3} b^{11} c^{4} d^{3} + 17920 \, a^{4} b^{9} c^{5} d^{3} - 57344 \, a^{5} b^{7} c^{6} d^{3} + 114688 \, a^{6} b^{5} c^{7} d^{3} - 131072 \, a^{7} b^{3} c^{8} d^{3} + 65536 \, a^{8} b c^{9} d^{3}}{b^{20} d^{4} - 40 \, a b^{18} c d^{4} + 720 \, a^{2} b^{16} c^{2} d^{4} - 7680 \, a^{3} b^{14} c^{3} d^{4} + 53760 \, a^{4} b^{12} c^{4} d^{4} - 258048 \, a^{5} b^{10} c^{5} d^{4} + 860160 \, a^{6} b^{8} c^{6} d^{4} - 1966080 \, a^{7} b^{6} c^{7} d^{4} + 2949120 \, a^{8} b^{4} c^{8} d^{4} - 2621440 \, a^{9} b^{2} c^{9} d^{4} + 1048576 \, a^{10} c^{10} d^{4}}\right )} x - \frac {b^{18} d^{3} - 48 \, a b^{16} c d^{3} + 960 \, a^{2} b^{14} c^{2} d^{3} - 10752 \, a^{3} b^{12} c^{3} d^{3} + 75264 \, a^{4} b^{10} c^{4} d^{3} - 344064 \, a^{5} b^{8} c^{5} d^{3} + 1032192 \, a^{6} b^{6} c^{6} d^{3} - 1966080 \, a^{7} b^{4} c^{7} d^{3} + 2162688 \, a^{8} b^{2} c^{8} d^{3} - 1048576 \, a^{9} c^{9} d^{3}}{b^{20} d^{4} - 40 \, a b^{18} c d^{4} + 720 \, a^{2} b^{16} c^{2} d^{4} - 7680 \, a^{3} b^{14} c^{3} d^{4} + 53760 \, a^{4} b^{12} c^{4} d^{4} - 258048 \, a^{5} b^{10} c^{5} d^{4} + 860160 \, a^{6} b^{8} c^{6} d^{4} - 1966080 \, a^{7} b^{6} c^{7} d^{4} + 2949120 \, a^{8} b^{4} c^{8} d^{4} - 2621440 \, a^{9} b^{2} c^{9} d^{4} + 1048576 \, a^{10} c^{10} d^{4}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \]

[In]

integrate(1/(2*c*d*x+b*d)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")

[Out]

32*c^2*arctan(-(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c + b*sqrt(c))/sqrt(b^2*c - 4*a*c^2))/((b^4*d - 8*a*b^2*
c*d + 16*a^2*c^2*d)*sqrt(b^2*c - 4*a*c^2)) + 2/3*(12*((b^16*c^2*d^3 - 32*a*b^14*c^3*d^3 + 448*a^2*b^12*c^4*d^3
- 3584*a^3*b^10*c^5*d^3 + 17920*a^4*b^8*c^6*d^3 - 57344*a^5*b^6*c^7*d^3 + 114688*a^6*b^4*c^8*d^3 - 131072*a^7
*b^2*c^9*d^3 + 65536*a^8*c^10*d^3)*x/(b^20*d^4 - 40*a*b^18*c*d^4 + 720*a^2*b^16*c^2*d^4 - 7680*a^3*b^14*c^3*d^
4 + 53760*a^4*b^12*c^4*d^4 - 258048*a^5*b^10*c^5*d^4 + 860160*a^6*b^8*c^6*d^4 - 1966080*a^7*b^6*c^7*d^4 + 2949
120*a^8*b^4*c^8*d^4 - 2621440*a^9*b^2*c^9*d^4 + 1048576*a^10*c^10*d^4) + (b^17*c*d^3 - 32*a*b^15*c^2*d^3 + 448
*a^2*b^13*c^3*d^3 - 3584*a^3*b^11*c^4*d^3 + 17920*a^4*b^9*c^5*d^3 - 57344*a^5*b^7*c^6*d^3 + 114688*a^6*b^5*c^7
*d^3 - 131072*a^7*b^3*c^8*d^3 + 65536*a^8*b*c^9*d^3)/(b^20*d^4 - 40*a*b^18*c*d^4 + 720*a^2*b^16*c^2*d^4 - 7680
*a^3*b^14*c^3*d^4 + 53760*a^4*b^12*c^4*d^4 - 258048*a^5*b^10*c^5*d^4 + 860160*a^6*b^8*c^6*d^4 - 1966080*a^7*b^
6*c^7*d^4 + 2949120*a^8*b^4*c^8*d^4 - 2621440*a^9*b^2*c^9*d^4 + 1048576*a^10*c^10*d^4))*x - (b^18*d^3 - 48*a*b
^16*c*d^3 + 960*a^2*b^14*c^2*d^3 - 10752*a^3*b^12*c^3*d^3 + 75264*a^4*b^10*c^4*d^3 - 344064*a^5*b^8*c^5*d^3 +
1032192*a^6*b^6*c^6*d^3 - 1966080*a^7*b^4*c^7*d^3 + 2162688*a^8*b^2*c^8*d^3 - 1048576*a^9*c^9*d^3)/(b^20*d^4 -
40*a*b^18*c*d^4 + 720*a^2*b^16*c^2*d^4 - 7680*a^3*b^14*c^3*d^4 + 53760*a^4*b^12*c^4*d^4 - 258048*a^5*b^10*c^5
*d^4 + 860160*a^6*b^8*c^6*d^4 - 1966080*a^7*b^6*c^7*d^4 + 2949120*a^8*b^4*c^8*d^4 - 2621440*a^9*b^2*c^9*d^4 +
1048576*a^10*c^10*d^4))/(c*x^2 + b*x + a)^(3/2)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(b d+2 c d x) \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\left (b\,d+2\,c\,d\,x\right )\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]

[In]

int(1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(5/2)),x)

[Out]

int(1/((b*d + 2*c*d*x)*(a + b*x + c*x^2)^(5/2)), x)